Elliptic curve 136857.5-f4 over number field Q(−3)\Q(\sqrt{-3})

• Label: 2.0.3.1-136857.5-f4
• Base field: $\Q(\sqrt{-3})$
• Conductor norm: $136857$
• CM: no
• Base change: no
• Q-curve: no
• Torsion order: $4$
• Rank: $1$

Base field $\Q(\sqrt{-3})$

Generator $a$, with minimal polynomial $x^{2} - x + 1$; class number $1$.

Weierstrass equation

${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-359a+448\right){x}+117a-2482$

This is a global minimal model.

Mordell-Weil group structure

$\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(-16 a - 33 : -311 a + 286 : 1\right)$	$0.64464081968558465631832258086038608252$	$\infty$
$\left(-23 a + 2 : 11 a - 1 : 1\right)$	$0$	$2$
$\left(5 a + 2 : -3 a - 1 : 1\right)$	$0$	$2$

Invariants

Conductor:	$\frak{N}$	=	$(-392a+49)$	=	$(-2a+1)\cdot(-3a+1)^{2}\cdot(3a-2)^{2}\cdot(-5a+3)$
Conductor norm:	$N(\frak{N})$	=	$136857$	=	$3\cdot7^{2}\cdot7^{2}\cdot19$
Discriminant:	$\Delta$	=	$2855694177a+193767903$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(2855694177a+193767903)$	=	$(-2a+1)^{6}\cdot(-3a+1)^{6}\cdot(3a-2)^{10}\cdot(-5a+3)^{2}$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$8745877105071325569$	=	$3^{6}\cdot7^{6}\cdot7^{10}\cdot19^{2}$
j-invariant:	$j$	=	$-\frac{28971353771}{23402547} a + \frac{23274207584}{7800849}$
Endomorphism ring:	$\mathrm{End}(E)$	=	$\Z$
Geometric endomorphism ring:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z$    (no potential complex multiplication)
Sato-Tate group:	$\mathrm{ST}(E)$	=	$\mathrm{SU}(2)$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$1$
Mordell-Weil rank:	$r$	=	$1$
Regulator:	$\mathrm{Reg}(E/K)$	≈	$0.64464081968558465631832258086038608252$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$1.28928163937116931263664516172077216504$
Global period:	$\Omega(E/K)$	≈	$0.710173400857236464901133312800346587840$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$192$  =  $( 2 \cdot 3 )\cdot2^{2}\cdot2^{2}\cdot2$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$4$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$6.3435565919467231800220834450581073718$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\begin{aligned}6.343556592 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 0.710173 \cdot 1.289282 \cdot 192 } { {4^2 \cdot 1.732051} } \\ & \approx 6.343556592 \end{aligned}$

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 4 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$	$N(\mathfrak{p})$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(-2a+1)$	$3$	$6$	$I_{6}$	Split multiplicative	$-1$	$1$	$6$	$6$
$(-3a+1)$	$7$	$4$	$I_0^{*}$	Additive	$-1$	$2$	$6$	$0$
$(3a-2)$	$7$	$4$	$I_{4}^{*}$	Additive	$-1$	$2$	$10$	$4$
$(-5a+3)$	$19$	$2$	$I_{2}$	Split multiplicative	$-1$	$1$	$2$	$2$

Galois Representations

The mod $p$ Galois Representation has maximal image for all primes $p < 1000$ except those listed.

prime	Image of Galois Representation
$2$	2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 136857.5-f consists of curves linked by isogenies of degrees dividing 8.

Base change

This elliptic curve is not a $\Q$-curve.

It is not the base change of an elliptic curve defined over any subfield.